73 research outputs found
The Radon transform and its dual for limits of symmetric spaces
The Radon transform and its dual are central objects in geometric analysis on
Riemannian symmetric spaces of the noncompact type. In this article we study
algebraic versions of those transforms on inductive limits of symmetric spaces.
In particular, we show that normalized versions exists on some spaces of
regular functions on the limit. We give a formula for the normalized transform
using integral kernels and relate them to limits of double fibration transforms
on spheres
Twistor Theory and Differential Equations
This is an elementary and self--contained review of twistor theory as a
geometric tool for solving non-linear differential equations. Solutions to
soliton equations like KdV, Tzitzeica, integrable chiral model, BPS monopole or
Sine-Gordon arise from holomorphic vector bundles over T\CP^1. A different
framework is provided for the dispersionless analogues of soliton equations,
like dispersionless KP or Toda system in 2+1 dimensions. Their
solutions correspond to deformations of (parts of) T\CP^1, and ultimately to
Einstein--Weyl curved geometries generalising the flat Minkowski space. A
number of exercises is included and the necessary facts about vector bundles
over the Riemann sphere are summarised in the Appendix.Comment: 23 Pages, 9 Figure
Self-dual gravity
Self-dual gravity is a diffeomorphism invariant theory in four dimensions that describes two propagating polarisations of the graviton and has a negative mass dimension coupling constant. Nevertheless, this theory is not only renormalisable but quantum finite, as we explain. We also collect various facts about self-dual gravity that are scattered across the literature
A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues
It is proved that a compact Kahler manifold whose Ricci tensor has two
distinct, constant, non-negative eigenvalues is locally the product of two
Kahler-Einstein manifolds. A stronger result is established for the case of
Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant
eigenvalues of the Ricci tensor are shown to exist in various situations: there
are homogeneous examples of any complex dimension n > 1, if one eigenvalue is
negative and the other positive or zero, and of any complex dimension n > 2, if
the both eigenvalues are negative; there are non-homogeneous examples of
complex dimension 2, if one of the eigenvalues is zero. The problem of
existence of Kahler metrics whose Ricci tensor has two distinct, constant
eigenvalues is related to the celebrated (still open) Goldberg conjecture.
Consequently, the irreducible homogeneous examples with negative eigenvalues
give rise to complete, Einstein, strictly almost Kahler metrics of any even
real dimension greater than 4.Comment: 18 pages; final version; accepted for publication in International
Journal of Mathematic
Supersymmetric Extensions of Calogero--Moser--Sutherland like Models: Construction and Some Solutions
We introduce a new class of models for interacting particles. Our
construction is based on Jacobians for the radial coordinates on certain
superspaces. The resulting models contain two parameters determining the
strengths of the interactions. This extends and generalizes the models of the
Calogero--Moser--Sutherland type for interacting particles in ordinary spaces.
The latter ones are included in our models as special cases. Using results
which we obtained previously for spherical functions in superspaces, we obtain
various properties and some explicit forms for the solutions. We present
physical interpretations. Our models involve two kinds of interacting
particles. One of the models can be viewed as describing interacting electrons
in a lower and upper band of a one--dimensional semiconductor. Another model is
quasi--two--dimensional. Two kinds of particles are confined to two different
spatial directions, the interaction contains dipole--dipole or tensor forces.Comment: 21 pages, 4 figure
Effect of short-range electron correlations in dynamic transport in a Luttinger liquid
The density operator in the Luttinger model consists of two components, one
of which describes long-wave fluctuations and the other is related to the rapid
oscillations of the charge-density-wave (CDW) type, caused by short-range
electron correlations. It is commonly believed that the conductance is
determined by the long-wave component. The CDW component is considered only
when an impurity is present. We investigate the contribution of this component
to the dynamic density response of a Luttinger liquid free from impurities. We
show that the conventional form of the CDW density operator does not conserve
the number of particles in the system. We propose the corrected CDW density
operator devoid of this shortcoming and calculate the dissipative conductance
in the case when the one-dimensional conductor is locally disturbed by a
conducting probe. The contribution of the CDW component to conductance is found
to dominate over that of the long-wave component in the low-frequency regime.Comment: 6 pages, 4 figures; updated to the published versio
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
In this paper, we present a quantum algorithm for dynamic programming
approach for problems on directed acyclic graphs (DAGs). The running time of
the algorithm is , and the running time of the
best known deterministic algorithm is , where is the number of
vertices, is the number of vertices with at least one outgoing edge;
is the number of edges. We show that we can solve problems that use OR,
AND, NAND, MAX and MIN functions as the main transition steps. The approach is
useful for a couple of problems. One of them is computing a Boolean formula
that is represented by Zhegalkin polynomial, a Boolean circuit with shared
input and non-constant depth evaluating. Another two are the single source
longest paths search for weighted DAGs and the diameter search problem for
unweighted DAGs.Comment: UCNC2019 Conference pape
Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries
We briefly review the hierarchy for the hyper-K\"ahler equations and define a
notion of symmetry for solutions of this hierarchy. A four-dimensional
hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy
with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden
symmetry if it admits a certain Killing spinor. We show that if the hidden
symmetry is tri-holomorphic, then this is equivalent to requiring symmetry
along a higher time and the hidden symmetry determines a `twistor group' action
as introduced by Bielawski \cite{B00}. This leads to a construction for the
solution to the hierarchy in terms of linear equations and variants of the
generalised Legendre transform for the hyper-K\"ahler metric itself given by
Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of
hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These
metrics are in this sense analogous to the 'finite gap' solutions in soliton
theory. Finally we extend the concept of a hierarchy from that of \cite{DM00}
for the four-dimensional hyper-K\"ahler equations to a generalisation of the
conformal anti-self-duality equations and briefly discuss hidden symmetries for
these equations.Comment: Final version. To appear in the August 2003 special issue of JMP on
`Integrability, Topological Solitons, and Beyond
Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
The paper contains a simple approach to reconstruction in Thermoacoustic and
Photoacoustic Tomography. The technique works for any geometry of point
detectors placement and for variable sound speed satisfying a non-trapping
condition. A uniqueness of reconstruction result is also obtained
Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism
The integral representations for the eigenfunctions of particle quantum
open and periodic Toda chains are constructed in the framework of Quantum
Inverse Scattering Method (QISM). Both periodic and open -particle solutions
have essentially the same structure being written as a generalized Fourier
transform over the eigenfunctions of the particle open Toda chain with
the kernels satisfying to the Baxter equations of the second and first order
respectively. In the latter case this leads to recurrent relations which result
to representation of the Mellin-Barnes type for solutions of an open chain. As
byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra
function in the case of GL(N,\RR) group.Comment: Latex+amssymb.sty, 14 page
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